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Theoretical Study of the Spectral and Charge-Transport Parameters of an Electron-Transporting Material Bis(10-hydroxybenzo[h]qinolinato)beryllium (Bebq2) Alexandra Ya. Freidzon,*,†,‡ Andrei A. Safonov,† and Alexander A. Bagaturyants†,‡ †

Photochemistry Center, Russian Academy of Sciences, Moscow 119421, Russia National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoye shosse 31, Moscow 115409, Russia

‡

S Supporting Information *

ABSTRACT: The multireference XMCQDPT2/CASSCF method is used to get insight into the charge transport mechanism of bis(10hydroxybenzo[h]quinolinato)beryllium (Bebq2) and to explain some features of its light absorption and emission in monomeric and dimeric forms. Energy profiles corresponding to electron and hole hopping in Bebq2 monomer and in three close-packed dimers that can occur in the solid phase are calculated. Our calculation revealed that charges and excitons could be either localized on individual ligands or delocalized over a pair of stacking ligands in dimers. Delocalized hole states serve as deep charge traps hindering hole transport. On the other hand, the electron states are localized, and hopping electron transport can take place with low barriers. The excited states of dimers exhibit exciton splitting. In some dimers, the transition dipole moment arrangement is unfavorable for luminescence. Therefore, our calculations explain why Bebq2 is an electron transporter (hole blocker) and why regular packing with ligand stacking in Bebq2 layers favors electron transport along the stacks but decreases luminescence.

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INTRODUCTION Charge transport in small-molecule disordered organic semiconductors proceeds via the hopping mechanism.1 In the hopping process, the charge is transferred from an initial state localized on a certain molecule (A) to a final state localized on a neighboring one (B).

It follows from the above that an accurate simulation of structural changes upon ionization and excitation of organic semiconductors is of crucial importance for predictive modeling of organic electronic devices. Hopping charge transport in disordered small-molecule semiconductors is usually described by the Marcus model.4−9 This process can be considered as electron exchange between two molecules, a neutral one and its radical anion (for electron transport) or radical cation (for hole transport). The rate constant of this process is

A* + B → A + B* Charge localization is accompanied by molecular reorganization (electron−phonon coupling). Therefore, the potential energy surface (PES) of the ionic supermolecule (AB)* is characterized by distinct minima separated by a barrier. Such charge localization accompanied by structure deformation can be considered as a manifestation of the pseudo-Jahn−Teller effect.2 The same also relates to excitation transfer; that is, the star may designate either a charged or excited state. Therefore, charge localization directly affects the kinetics of charge and exciton transport in organic semiconductors. Another issue in organic electronics is chemical stability of the materials during device operation.3 The molecular structure changes when charges or excitons travel through the material, causing weakening of some bonds (which can further break) or electron density accumulation (or depletion) on some atoms, resulting in the formation of highly reactive species. These structural changes are also directly related with charge or exciton localization in the molecule. © 2015 American Chemical Society

ket =

2π |HAB|2 ℏ

⎛ (λ + ΔG°)2 ⎞ 1 ⎟ exp⎜ − 4λk bT ⎠ 4πλk bT ⎝

where ket is the rate constant for electron transfer, kb is the Boltzmann constant, T is the absolute temperature, ΔG0 is the Gibbs free energy of the electron transfer reaction (or site energy disorder), λ is the reorganization energy of the molecule, and |HAB| is the so-called hopping integral. For hopping between molecules of same sort (A = B), ΔG0 ≈ 0. All these parameters, ΔG0, λ, and |HAB| depends on the layer material and can be calculated ab initio. Received: August 24, 2015 Revised: November 3, 2015 Published: November 9, 2015 26817

DOI: 10.1021/acs.jpcc.5b08239 J. Phys. Chem. C 2015, 119, 26817−26827

Article

The Journal of Physical Chemistry C Many OLED materials form amorphous layers. In some cases, local or low-dimensional ordering is observed, depending on the layer deposition conditions. This ordering may have either advantageous or adverse effects on the layer performance. (Bis(10-hydroxybenzo[h]quinolinato)beryllium (Bebq2) is a promising electron-transporting and hole-blocking, 10,11 host,12−14 and electroluminescent15,16 material in organic light emitting devices (OLEDs). Its electroluminescence (EL) properties were shown to be superior to those of tris(8hydroxyquinolinato)aluminum (Alq3). Bebq2 emits in the blue-green region, which is especially valuable for OLED applications.17 Studies of the effect of the film deposition rate on the performance of Bebq2-based OLEDs18,19 show that at lower deposition rates the photoluminescence (PL) efficiency significantly decreases and the electron mobility increases. This can be attributed to the fact that ordered Bebq2 aggregates are easier formed at low deposition rates. Hence, the chargetransport and light-emission properties of Bebq2 layers strongly depend on the mutual arrangement of Bebq2 molecules in the layers. Bebq2 (Figure 1) contains two equivalent ligands, and the charge or excitation in such complexes can be either delocalized

Figure 2. Stacking chains in (a) Bempp2 (bis(2-(4-methylpyridin-2yl)phenolato)-beryllium, CCDC 827923) and (b) Bepp2 (bis(2-(2Hydroxyphenyl)pyridine)-beryllium, CCDC 147265).23,24

consider a larger series of Bebq2 dimers with different ligand arrangements. In addition, we study the excited states of Bebq2 monomer and dimers and demonstrate that excitons can be either localized or delocalized, depending on the molecular packing, and exciton localization (or delocalization) influences the luminescence efficiency. We explain why Bebq2 is an electron transporter and hole blocker and how molecular packing can affect the performance of Bebq2 as an emitter or electron transporter. Because of the chemical similarity of Bebq2 and Alq3, the conclusions on the mechanism can be extended to explain the peculiarities of charge transport and luminescence of Alq3.

Figure 1. Chemical structure of the Bebq2 molecule.

or localized on any of them (see, for example, refs 20 and21). Hence, there are two possibilities for charge localization in the monomer and four possibilities for charge localization in dimers. Since the ligands are equivalent, the localized states are quasi-degenerate and undergo a pseudo-Jahn−Teller distortion.2 Previously, we have demonstrated22 that the electron or hole in the charged Bebq2 monomers and dimers is localized on one of the hydroxybenzoquinolinate ligands. Therefore, the charge transport process in this complex involves not only intermolecular but also intramolecular hopping steps. We have also shown that this localization is accompanied by a notable structural reorganization. However, our previous calculations could not explain why Bebq2 is an efficient electron transporter and hole blocker. To elucidate this, it is necessary to consider the packing of Bebq2 in OLED layers. There is no crystal structure data for Bebq2, but these data are available for similar complexes with 2-(2-hydroxyphenyl)pyridine (pp) and 2-(2-hydroxyphenyl)-4-methyl-pyridine (mpp).23,24 The crystal packing of these complexes exhibits chains of stacked molecules (Figure 2). One can suggest that it is the stacking contacts of this type that are responsible for charge hopping in Bebq2. In this case, charge transport is a sequence of intra- and intermolecular hopping acts from one 10-hydroxybenzo[h]quinolinate fragment to another. In this work, we focus on the accurate ab initio calculation of some important transport and spectral parameters of Bebq2. We use a larger active space than in ref 22 and thoroughly

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COMPUTATIONAL DETAILS Eqs 1−4 of Appendix show that the reorganization energy λ can be derived directly from atomistic quantum chemical calculations. As for the hopping integral |HAB|, its calculation is rather complicated and involves at least two electronic states and three points on the PES of the ionic supermolecule (initial state, final state, and barrier top). At the same time, |HAB| is frequently calculated as an energy splitting in dimer (ESD) [i.e., a HOMO (for holes) or LUMO (for electrons) splitting in the neutral dimer at its equilibrium geometry (see, for example, refs 25−32)]. This approximation is rather oversimplified, but it clearly reveals both the trends in series of compounds and the dependence of the hopping integral on the distance and orientation of the monomers in the dimer. As mentioned above, accurate calculation of |HAB| should be performed at the avoided crossing point and, therefore, requires multireference methods. Hence, we optimized the geometries by a state-specific CASSCF method (SS-CASSCF) with Grimme’s dispersion correction33−35 adjusted for the Hartree− Fock method because no dispersion correction is implemented for CASSCF. To facilitate geometry optimization, we started with ROHF-D (i.e., ROHF with the same Grimme’s dispersion correction) optimized structures, which proved to be a very good starting approximation for SS-CASSCF-D optimization. 26818


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The Journal of Physical Chemistry C

Figure 3. Hopping pathways in Bebq2 dimers. Green arrows show intramolecular hopping; orange arrows show intermolecular hopping.

and charge hopping pathways in them. Intermolecular hopping denoted by orange arrows can occur between stacking ligands, and intramolecular hopping denoted by green arrows occurs between ligands bound to the same central ion. Dimers of this type frequently occur in Bebq2 layers built by molecular dynamics.39

The energies of the electronically excited states and ionic species were calculated by the state-averaged XMCQDPT/SACASSCF method.36 The active space included one HOMO and one LUMO from each ligand and the number of electrons corresponding to the target (neutral, cationic, or anionic) state. Thus, the active space was (4e,4o) for neutral monomers, (3e,4o) for monomeric cations, and (5e,4o) for monomeric anions. Similarly, for dimers, the active spaces were (8e,8o), (7e,8o), and (9e,8o), respectively. State averaging was performed over equivalent states. For ionic species, these are the states corresponding to the charge localization on each ligand. For neutral states, these are singly excited triplet or singlet states originating from one-electron excitation from each ligand HOMO to each ligand LUMO. For singlets, the ground state was also included. Therefore, for monomers state averaging was SA(2) for ions, SA(3) for neutral singlets, and SA(2) for triplets. (In this case, charge-transfer states were excluded from state averaging but not from the Hamiltonian.) For dimers, it was SA(4) for ions and SA(17) for neutral singlets. The Firefly program package37 was used throughout the calculations. All the calculations were performed using splitvalence polarized double-ζ basis set.38 If the two monomers A and B are of the same sort, the barrier top (or an avoided crossing point, see Appendix) can be approximated by the midpoint on the reaction path from the minimum corresponding to the charge localized on A to the minimum corresponding to the charge localized on B. (This is exactly fulfilled if A and B are symmetry related.) In this geometry, the charge is expected to be equally localized (or delocalized) over the two monomers. The midpoint geometries were taken as an average of the internal coordinates of the corresponding ionic dimers. In our view, this approach is more accurate than the ESD method, which employs the geometry of the neutral dimer. We considered three tight dimers (denoted as Type A, Type B, and Type C) with ligand stacking. Figure 3 shows their structures

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RESULTS AND DISCUSSION Calculated Structures of Bebq2 Monomer and Dimers. The monomeric Bebq2 molecule has C2 symmetry. The calculated monomer geometry agrees with the experimental values observed in the related structures of Bempp2 and Bepp223,24 with the exception of Be−N bond, which is ∼0.08 Å longer than in these X-ray structures. This is not necessarily the calculation artifact; instead, this may reflect the difference between the coordination of hydroxybenzoquinoline and hydroxyphenylpyridine ligands to beryllium. Dimer A has no symmetry, although C2 might be expected. Dimer B has approximate C2 symmetry, and dimer C has Ci symmetry with sufficient accuracy. The distances between stacking ligands are 3.2−3.3 Å. Calculated Charge Hopping Parameters of Bebq2 Monomer. Previously, we have found that the charged states in the Bebq2 molecule are localized on individual ligands; hence, we can speak about intramolecular (interligand) charge transfer. In the hopping midpoint, the charge is delocalized over the two ligands, but the intramolecular hopping barriers are low for both electrons and holes. Table 1 shows the reorganization energies both for intramolecular and intermolecular hopping. The reorganization energies for the intramolecular hopping given in this table are calculated as level splittings at the end points of the energy profiles in Figure 4. For comparison, we estimated |HAB| for intramolecular charge hopping as ESD. To do this, we used Hartree−Fock orbitals of the optimized structures of Bebq2 monomer. Its frontier orbitals are present in pairs [i.e., each orbital is 26819


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Therefore, the charge hopping in dimers A and C consists of charge capture from the neighboring molecule by an unpaired ligand L2, intramolecular hopping to L1, intermolecular hopping between stacked ligands L1 and L3, intramolecular hopping to the other unpaired ligand L4, and passing the charge to the next neighbor. If the molecules are packed in stacks similar to those observed in Bepp2 or Bempp2 crystals (Figure 2), the charge can travel along the chain. Regular chainlike packing of dimers B is hardly possible. However, in amorphous layers built in MD simulations39 such dimers can contact with other types of dimers or with monomers and form irregular stacks. Therefore, after capturing the charge in dimer B from the neighboring molecules, it may either hop from L2 to L1 to L3 to L4 and again to L2 or leave the dimer through hopping to another neighbor. The calculated hole hopping profiles for the three considered dimers are shown in Figure 5. One can see that the profiles for dimers A and C exhibit a deep minimum corresponding to charge delocalizaton between L1 and L3 in the midpoint for L1−L3 hopping. Such minima can serve as hole traps because hopping from them to L2 or L4 requires ∼0.5 eV to overcome the barriers. For dimer B, such minimum exists for the hole delocalized between L2 and L4, but it is rather shallow at ∼ 0.2 eV. The presence of such hole traps can explain why Bebq2 is a hole blocker. The calculated electron hopping profiles for the three considered dimers are shown in Figure 6. Here we can see that the minima correspond to the electron localized on individual ligands, while delocalized states corresponding to the midpoints are barrier tops. Therefore, the electrons in these dimers undergo hopping transport. Dimer A demonstrates a usual electron hopping profile with relatively low (up to 0.35 eV) barriers. Intramolecular hopping integrals for L2−L1 and L3−L4 hopping are 0.09 and 0.12 eV, respectively. Intermolecular hopping integral for L1−L3 hopping is 0.29 eV. In dimer B, there are two minima corresponding to electron localization on L2 and L4. Hopping to L1 and L3 requires 0.3 eV, and hopping between L1 and L3 is almost barrierless. Intramolecular hopping integrals for L2−L1 and L3−L4 hopping are 0.11 eV. Intermolecular hopping integrals are 0.49 eV for L1−L3 hopping and 0.04 eV for L2−L4 hopping. The barrier for this last step is 0.42 eV. In dimer C, intramolecular hopping integrals for L2−L1 and L3−L4 hopping are 0.09 eV. As for intermolecular L1−L3 hopping, the midpoint corresponding to this trajectory on the

Table 1. Calculated Inter- And Intramolecular Reorganization Energies and Intramolecular Hopping Integrals λholeinter (eV)

λelinter (eV)

λholeintra (eV)

λelintra (eV)

|HAB|holeintra (eV)

|HAB|elintra (eV)

0.39

0.63

0.62

0.73

0.08

0.05

represented by either a sum or difference of the orbitals localized on the two ligands: Ψ1 = φA + φB; Ψ2 = φA − φB (Figure S1)]. The energy difference between HOMO and HOMO−1 is 0.11 eV, the energy difference between LUMO and LUMO+1 is 0.07 eV. Therefore, |HAB|hole is 0.05 eV and |HAB|el is 0.03 eV. This agrees with the values of Table 1 calculated directly. In addition, we calculated the intramolecular reorganization energies of ionization in the R(O)HF/SVP approximation using displaced multimode harmonic oscillator approximation, also known as the Lax model40,41 implemented in the KTS program.42−45 In this approximation, λholeintra = 0.41 eV, λelintra = 0.43 eV, which also agrees with the values of Table 1. These calculations showed that the most important reorganization modes of both electron attachment and detachment include in-plane deformations of the ligands and migration of Be2+ from the center of the complex toward the ligand bearing the electron or from the ligand bearing the hole. Direct estimation of the intramolecular reorganization energy by eqs 1 and 2 (see Appendix) is somewhat more involved. In this case, it is ligand rather than entire complex that undergoes reorganization; therefore, the reorganization energy should be calculated for the free ligand in its protonated form bqH, because H mimicks Be ion in the complex. Thus, calculated reorganization energies are λholeintra = 0.48 eV and λelintra = 0.45 eV, which also agrees with the values of Table 1 taking into account the difference between the free bqH ligand and bq− anion in the Bebq2 complex and excellently agrees with the values calculated within the Lax model. The reorganization energies and hopping integrals for the electron and hole hopping calculated for the Bebq2 monomer do not show any distinct trends to preferential electron hopping and, therefore, do not explain why this complex is an electron transporter and hole blocker. Therefore, intermolecular processes should be considered. Calculated Charge Hopping Parameters of Bebq2 Dimers. Figure 3 shows charge hopping pathways in the three tight dimers. Dimers A and C have only one pair of stacking ligands, while the ligands in dimer B are stacked pairwise.

Figure 4. Energy profiles (eV) for the intramolecular charge hopping in Bebq2 monomer (a) hole and (b) electron. Singly occupied orbitals are shown for the lowest-energy term. L1 is the left ligand; L2 is the right one. 26820


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Figure 5. Energy profile (eV) for hole hopping profiles in dimers (a) A, (b) B, and (c) C. Singly occupied orbitals are shown for the lowest energy term.

potential energy surface seems to be close to a conical intersection: the level splitting in this point is 0.03 eV. The barrier between the points corresponding to the electron localized on L1 and L3 is rather high at ∼0.9 eV. Obviously, for hopping to proceed adiabatically, there should be a bypass trajectory with lower barrier.

Indeed, if we compare the top views of the anionic structures with electron localized on L1 and L3 (we call these structures C-L1 and C-L3, respectively, Figure 7, panels a and b), we can see that direct electron transfer from one form to the other requires large-amplitude rotation of one or both monomeric Bebq2 units. However, we found two slightly higher (by 0.17−0.19 eV) 26821


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Figure 6. Energy profile (eV) for electron hopping in dimers (a) A, (b) B, and (c) C. Singly occupied orbitals are shown for the lowest energy term.

lying stationary point structures with electron localized on the same ligands (we call them C-L1a and C-L3a, respectively, Figure 7, panels c and d). These structures require only moderate reorganization without large-amplitude motion for the electron transfer from C-L1 to C-L3a and from C-L3 to C-L1a. We built an energy profile corresponding to the linear transit from C-L1 to C-L3a and from C-L3 to C-L1a (Figure 8). These trajectories bypass the conical intersection point. As expected, the electron localization changes from L1 to L3 or

vice versa with full electron delocalization in the midpoint. However, the highest points of these trajectories are C-L3a and C-L1a, rather than midpoints. Therefore, we may conclude that C-L3a and C-L1a are close to the transition states for the electron transfer from L1 to L3 or backward. The problem of large-amplitude motion in going from C-L3a to C-L3 and from C-L1a to C-L1 still remains. Such motion is hardly possible in closely packed crystals or melted-and-cooled layers but possible in loose amorphous layers. 26822


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Figure 7. Top view of the anionic forms of dimer C with electron localized (a and c) on L1 (C-L1 and C-L1a) and (b and d) on L3 (C-L3 and C-L3a). (a and b) Local minima C-L1 and C-L3, and (c and d) intermediate structures C-L1a and C-L3a.

Figure 8. Energy profile (eV) for electron hopping along the linear transit from C-L1 to C-L3a (dark blue and pink lines) and from C-L3 to C-L1a (green and cyan lines).

Calculated Optical Properties of Bebq2 Monomer. The calculated absorption and emission energies of Bebq2 monomer (Table 2, Figure 9) agree well with the experiment.14,15,46,47 Table 2. Calculated Spectral Parameters of Bebq2 Monomer Compared with Experimental Data calculated experimental a

Eex (eV)

ET (eV)

Efl (eV)

Eph (eV)

3.00 2.95−3.04a

2.55

2.15 2.42−2.56b

1.55 2.16c

Refs 46 and 47. bRefs 14, 15, and 47. cRef 47.

Vertical excitation of Bebq2 is delocalized. The absorption band consists of two quasi-degenerate transitions of comparable intensity. The emitting states, however, are localized on

Figure 9. Calculated absorption spectra of Bebq2 monomer compared with experimental data of refs 46 and 47. 26823


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the antiparallel arrangement of their dipoles in dimer C leads to Davydov splitting of the electronic transitions and in very low intensity of the first transition. In dimer A, the interaction between the stacking ligands is also strong; however, the antiparallel arrangement of their dipoles is not as pronounced as in dimer C, and the intensity of the first transition is higher. In dimer B, the dipoles of the ligands are arranged in parallel, and all the transitions are intensive. Low intensity of the first transition in dimer C means that the luminescence from this state also will be forbidden. This explains why regular packing of Bebq2 molecules achieved through slow deposition results in low luminescence efficiency.

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CONCLUSIONS Multireference methods are successfully applied to calculate the charge transport and spectral parameters of Bebq2 and to explain some features of charge transport and luminescence of this material. We demonstrated that charge transport in Bebq2 is a series of intra- and intermolecular hopping steps. In the monomer, the charge is completely localized on one of the ligands and the intramolecular hopping integral is small. In dimers, charge localization depends on the arrangement of the molecules. The hole tends to delocalize over the central ligand pair to form a hole trap. The excessive electron tends to localize on the peripheral ligands, and electron transport is a series of hops

Figure 10. Energy levels of Bebq2 monomer (excitation localization is shown by asterisk).

individual ligands, thus exhibiting a pseudo-Jahn−Teller effect (Figure 10). Calculated Optical Properties of Bebq2 Dimers. The calculated absorption energies of Bebq2 dimers are given in Table 3 and Figure 11. All the transitions are slightly red-shifted with respect to the monomer. In dimers A and B, the transitions are mostly local, while in dimer C, the two lowest transitions are delocalized over stacking ligands. Strong interaction between the central stacking ligands together with

Table 3. Calculated Spectra of the Four Lowest Singlet Transitions dimer A

dimer B

dimer C

λ (nm)

E, eV

f

λ (nm)

E (eV)

f

λ (nm)

E (eV)

f

460 456 447 442

2.693 2.721 2.772 2.808

0.107 0.485 0.627 0.918

462 459 448 445

2.686 2.702 2.767 2.788

0.461 0.464 0.719 0.563

465 453 442 437

2.667 2.738 2.807 2.835

0.015 0.799 0.000 1.325

Figure 11. Calculated absorption spectra of the three dimers (stick spectra with Gaussian broadening with fwhm = 0.05 eV). 26824


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the ionization potential and electron affinity of the neutral and charged molecule. Thus, for hole hopping

with relatively small barriers. This explains the electrontransporting and hole-blocking properties of Bebq2. We have also shown that stacking favors intermolecular charge transfer, but the fluorescence of the stacked molecules can be forbidden due to antiparallel arrangement of the ligands.

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λhole = IP(A0) − EA(A+)

(1)

0

where IP(A ) is the ionization potential of the neutral molecule IP(A0) = E(A+//A0) − E(A0//A0), EA(A+) is the electron affinity of the cation EA(A+) = E(A+//A+) − E(A0//A+), A0// A0 is the neutral molecule in its optimized geometry, A+//A0 is the cation in the geometry of the neutral molecule, A+//A+ is the cation in its optimized geometry, and A0//A+ is the neutral molecule in the geometry of the cation. Similarly, for electron hopping,

APPENDIX

Marcus Theory for Charge Transport: A Relationship between Atomistic and Charge-Hopping Parameters

To better understand which atomistic parameters govern the conductivity of organic semiconductors, let us consider charge transport in disordered small-molecule organic semiconductors in detail. The reorganizaion energy λ is the energy dissipated when a system that has undergone “vertical” electron transfer relaxes to the equilibrium geometry for its new charge distribution. Consider a neutral molecule with its PES and a charged molecule with its PES (Figure 12). When the charge is transferred from the charged molecule to its neutral counterpart, the neutral molecule relaxes to the equilibrium state. At the same time, its charged counterpart relaxes to the equilibrium state. The total energy of relaxation is the reorganization energy. It can be calculated as the difference between

λelectron = IP(A−) − EA(A0)

(2)

−

where IP(A ) is the ionization potential of the anion and EA(A0) is the electron affinity of the neutral molecule. Alternatively, if we regroup the terms in the expression for the reorganization energy (1), it can be expressed as the vertical excitation energy of the dimer (Figure 13) consisting of two noninteracting monomers to the state where the charge has swapped, but the geometry has not yet relaxed. Indeed, if we neglect the intermolecular interaction in the dimer, λhole = IP(A0) − EA(A+) = [E(A+//A0) − E(A0 //A0)] − [E(A+//A+) − E(A0 //A+)] = [E(A+//A0) − E(A+//A+)] − [E(A0 //A0) − E(A0 //A+)] = E[(A0A+)*] − E(A0A+), (3)

and similarly for the λelectron. λelectron = E[(A0A−)*] − E(A0A−)

(4)

Hence, the difference in λ calculated for the monomer by eqs 1 or 2 and in a real dimer by eqs 3 or 4 can indicate the degree of molecular polarization in the dimer. The hopping integral is the electronic coupling between the initial and final states of a bimolecular system sharing one charge carrier (electron or hole). The barrier top on the charge transfer path is the point of an avoided crossing. For a onedimensional two-state model (Figure 13), the wavefunction of

Figure 12. Single-molecule parameters needed to calculate the reorganization energy λ for hole hopping. Dark dots denote the hole.

Figure 13. Reorganization energy and hopping integral as bimolecular parameters. Dark dot denotes the charge carrier. 26825


The Journal of Physical Chemistry C any state can be expressed as a linear combination of the initial and final states

The eigenvalue problem is the following: ⎡ HAA − E HAB ⎤⎡ cA ⎤ ⎢ ⎥⎢ ⎥ = 0, where HBB − E ⎦⎣ c B ⎦ ⎣ HBA (5)

HBB = ⟨φB|H |φB⟩

(6)

HAB = ⟨φA |H |φB⟩ = HBA

(7)

EB =

(HAA + HBB) −

(HAA − HBB)2 + 4HAB 2 2

(HAA + HBB) +

(8)

(HAA − HBB)2 + 4HAB 2 2

(9)

Naturally, |HAB| depends on the arrangement of the molecules in the dimer (or on the molecular packing in the material), and therefore, this is the factor that governs the charge transport in the layer.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.5b08239. Cartesian coordinates of the calculated Bebq2 monomer and three dimers (types A, B, and C). Frontier Hartree− Fock orbitals of Bebq2 monomer. Dominant configurations in the lowest excitations of the three Bebq2 dimers (PDF)

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ABBREVIATIONS

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REFERENCES

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The splitting of the levels in the avoided crossing point is twice the interaction |HAB| (eq 7) between the initial and final states. The larger |HAB|, the lower the hopping barrier. EA =

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Bebq2, bis(10-hydroxybenzo[h]qinolinato)beryllium; Bempp2, (bis(2-(4-methylpyridin-2-yl)phenolato)-beryllium); Bepp2, (bis(2-(2-Hydroxyphenyl)pyridine)-beryllium); CASSCF, complete active space self-consistent field; XMCQDPT2, secondorder extended multiconfigurational quasidegenerate perturbation theory

Ψ = cAφA + c BφB

HAA = ⟨φA |H |φA ⟩

Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank S.V. Emelyanova (NRNU MEPhI, Photochemistry Center) and Dr. A.V. Odinokov (Photochemistry Center) for providing us with unpublished data on MD simulations of Bebq2 layers. The calculations were performed using the facilities of the Joint Supercomputer Center of Russian Academy of Sciences and the Supercomputing Center of Lomonosov, Moscow State University.48 This work was supported by the Russian Science Foundation (Grant 14-4300052). 26826


Article

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